Amélia Fonseca

Nonzero star products

Many authors have considered the problem of determining necessary and sufficient conditions for the star product of mvectors to be zero. Probably the most remarkable results are those of Merris [9] and Gamas [4]. Our purpose is to generalize these two results to results arbitray symmetry classes of tensors.

On the multilinearity partition of an irreducible character

The concept of k-dimension of a family of vectors is introduced. Using this concept we are able to define the k-index of a symmetry class of tensors, generalizing the Marcus—Chollet index. Both notions enable us to associate to each irreducible character of a subgroup G of Sm a partition of m. It is also proved that if G=Sm this partition coincides with the partition usually associated with the irreducible characters of Sm .

The covering number of the elements of a matroid and generalized matrix functions

Abstract Let S be a nonempty finite set with cardinality m . Let M be a matroid on S with no loops. The covering number of an element x in S is the smallest positive integer k such that x is a coloop of the union of k copies of M . We investigate connections between the structure of M and the values of the covering numbers of elements of S . Applications to the study of the rank partition and generalized matrix functions are presented.

ON THE EQUALITY OF FAMILIES OF DECOMPOSABLE SYMMETRIZED TENSORS

Abstract We give some necessary and equivalent conditions for x σ(1) ∗⋯∗x σ(m) =y σ(1) ∗⋯∗y σ(m) to hold, for all σ ∈ S m , in arbitrary symmetry classes of tensors.

The r -depth of a matroid

Abstract We introduce the concept of depth and r -depth of a matroid M , proving that the sequence of the r -depths is the conjugate partition of the rank partition of M . The notion of quasi-transversal is defined and its properties stated. We also present connections between the concept of r -depth, the quasi-transversals of M and the circuits of the k th power of M .

Symmetry classes of tensors:dual indices

In this article we use the concept of r-depth to introduce the notion of dual multilinearity index. Using this concept we associate to each irreducible character of a subgroup G of Sm an improper partition of m. It is proved that of G = Sm this partition is the partition usually associated to the irreducible characters of Sm .

Multilinearity partitions of characters of embedded groups

In [2] a partition of m was associated to each irreducible C character λ of a subgroup G of Sm ; this partition was called the multilinearity partition of λ. When G = Sm the multilinearity partition of λ coincides with the partition usually associated to λ. Suppose now that p is an integer, 1 ≤ p < m, and G stabilizes the points p + l,…,m. in these conditions we can see G as a subgroup of Sp so, if λ is an irreducible C-character of G, it is possible to associate to λ a multilinearity partition which is a partition of p and a multiiinearity partition which is a partition of m. What is the connection between these two partitions? The article answers this question.

Computing the rank partition and the flag transversal of a matroid

Abstract Based only on the knowledge of the values of the rank function two procedures are given to compute the rank partition of a matroid. It is also shown that one of these procedures enables one to determine the flag transversal of a matroid also.

Families of vectors with prescribed rank partition and a prescribed subfamily

Abstract We investigate the conditions under which there exists a family of vectors with prescribed rank partition and a prescribed subfamily.

COLORABILITY OF INDUCED MATROIDS

Abstract Let G be a bipartite graph with bipartition {X, Y} such that there is a matching which matches Y with a subset of X. Let M be a matroid on X, and for each positive integer k, let M (k) be the matroid on X which is the union of M with itself k times. The G-induced chromatic number of M is the smallest integer p such that Y is an independent set of the induced matroid ( M (p))G. This chromatic number gives rise to a partition of |Y| called the G-induced chromatic partition. This partition majorizes partitions corresponding to other induced partitions, of Y.